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The Bakry-´Emery Ricci Tensor: Application to Mass Distribution in Space-Time

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The Bakry-´Emery Ricci Tensor: Application to Mass Distribution in Space-Time The Bakry-Emery´ Ricci tensor: Application to mass distribution in space-time Ghodratallah Fasihi-Ramandia∗ Department of Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran Abstract The Bakry-Emery´ Ricci tensor gives an analogue of the Ricci tensor for a Riemannian manifold with a smooth function. This notion motivates new version of Einstein field equation in which mass becomes part of geometry. This new field equation is purely geometric and is obtained from an action principle which is formed naturally by scalar curvature associated to the Bakry-Emery´ Ricci tensor. Keywords: Bakry-Emery´ Ricci, Mass distribution, Space-time, Einstein field equation. 1 Introduction No well-defined current physical theory claims to model all nature; each intentionally neglects some effects. Roughly, general relativity is a model of nature, especially of gravity, that neglects quan- tum effects. In fact, general relativity describes gravity in the context of 4-dimensional Lorentzian manifolds by Einstein field equation. Einstein argued that the stress-energy of matter and electro- magnetism influences space-time (M,g) and suggested his field equation as follows [7]. 1 Ric − Rg = T + E, (1.1) 2 arXiv:1902.04428v1 [math.GM] 10 Feb 2019 where, T and E are stress-energy tensor of matter and electromagnetic field, respectively. The expressions in the right hand side of the Einstein equation 1.1 are physical concepts whilst the left hand side is completely geometric. In fact, influence of matter and electromagnetism on space-time is not described directly by the geometry of space-time manifold (M,g). Hence, a question naturally arises here, is: how to geometrize matter and electromagnetism? Attempts at geometrization of electromagnetism also known as unification of gravity and elec- tromagnetism have been made ever since the advent of general theory of relativity. Most of these attempts share the idea that Einstein’s original theory must in some way be generalized that some part of geometry describes electromagnetism. Nevertheless, there are relatively few results on ge- ometrization of mass in general relativity or, more generally applicable to space-time manifold, and ∗[email protected] (Gh. Fasihi Ramandi) 1 many of them rely on dimension 5 or other geometric structures like Lie algebroids (see [5], [6] and [2]). In the following, we present some clues which show that replacing the Ricci curvature tensor of a space-time manifold (M,g) by the Bakry-Emery´ Ricci tensor provides us a good apparatus for geometrization of mass. 1 (1) Because of the Einstein equation Ric− Rg = T +E, any attempt on geometrization of mass 2 and electromagnetism then becomes a condition on the Einstein equation and so, ultimately, on the Ricci curvature. Once one has a condition on Ricci curvature, one has a tool to geometrization of mass. Suppose that f is a smooth function on a Riemannian manifold (M,g), then the m-Bakry-Emery´ Ricci tensor is defined as 1 Ricm = Ric + Hessf − df ⊗ df, f m where, Ric stands for Ricci curvature tensor of (M,g) and m is a non-zero constant that is also ∞ allowed to be infinite, in which case we write Ricf = Ric + Hessf. The Ricci curvature tensor can describes gravity and the m-Bakry-Emery´ Ricci tensor is capable of describing gravity and matter, simultaneously. In fact, the Ricci curvature describes gravity and f can play the role of potential for mass distribution in space-time. (2) From the point of view of the foundation of general relativity, projective structures play an important role; the reason relies on the relations ’free falling (massive) particles-pre-geodesics- projective structures’ (see [4]). Let ∇ be the Levi-Civita connection of a Riemannian manifold (M,g) of dimension n. For a 1-form α define α ∇X Y = ∇X Y − α(X)Y − α(Y )X. The affine connection ∇α is torsion-free, moreover it is projectively equivalent to ∇, meaning that ∇α has the same geodesics, up to re-parametrization. It can be seen that any torsion-free connection projectively equivalent to ∇ is of the form ∇α for some α. df To see the link to Bakry-Emery´ Ricci curvature, let α = , then we recover a Bakry-Emery´ n − 1 Ricci Ricci tensor. As a simple calculation shows that (Proposition 3.3 in [9]) α df ⊗ df Ric∇ = Ric + Hessf + = Ric1−n. g n − 1 f In other words, the Ricci tensor of a projectively equivalent connection is exactly the (1−n)-Bakry- Emery´ Ricci tensor. The Ricci tensor of a projectively equivalent connection can be seen as a condition on Ricci curvature and so can be a tool for geimetrization of mass. (3) We already know Einstein metrics play a crucial role in general theory of relativity. Hence, the key is provided by generalization of Einstein metrics should probably capable of solve our concern. 2 A triple (M,g,f) (a Riemannian manifold (M,g) with a smooth function f on M) is said to be (m−)quasi-Einstein if it satisfies the equation 1 Ricm = Ric + Hessf − df ⊗ df = λg, f m for some λ ∈ R. The above equation is especially interesting in that when m = ∞ it is exactly the gradient Ricci soliton equation; when m is a positive integer, it corresponds to warped product Einstein metrics (see [3]); when f is constant, it gives the Einstein equation. We call a quasi-Einstein metric trivial when f is constant. Following the terminology of Ricci solitons, a quasi-Einstein metric on a manifold M will be called expanding, steady or shrinking, respectively, if λ < 0, λ = 0 or λ> 0. As we have noticed, the Bakry-Emery´ Ricci tensor is related to notion of quasi Einstein metrics. So, we hope Bakry-Emery´ Ricci tensors provide a good apparatus for geometrization of mass in the next section. 2 New Field Equation In this section, we consider (M,g) as a space-time manifold whose dimension is n and 2 ≤ n. Each smooth function f on M determines a 1-Bakry-Emery´ Ricci tensor on M. Let M denotes the set of all 1-Bakry-Emery´ Ricci tensor on a manifold M then, M can be written as the set of some pairs (g, f), where g is a semi-Riemannian metric and, f is a smooth function on M. In fact, M can be consider as an open subset of an infinite dimensional vector space. Denote canonical volume form of a meter g on the oriented manifold M by dVg. It is well-known that critical points of Einstein-Hilbert functional L(g)= RgdVg, ZM are metrics which satisfy the following equation, 1 Ric − Rg =0. 2 The above equation is Einstein field equation in vacuum. It motivates to define a new Hilbert- Einstein functional L : M→ R L(g, f)= R(g,f)dVg, ZM 1 1 where, R(g,f) = tr(Ricf ) is called scalar curvature of Ricf . Equations in which the critical points of this functional satisfy will be called the new field equation. For a symmetric 2-covariant tensor s on M, and a smooth function h set, g˜(t)= g + ts f˜(t)= f + th 3 For sufficiently small t, (˜g(t), f˜(t)) is a 1-Bakry-Emery´ Ricci tensor on M and is a variation of (g, f). The Bakry-Emery´ Ricci tensor (g, f) is a critical 1-Bakry-Emery´ Ricci tensor for Hilbert action if and only if for any pair (s, h): d d | L g˜(t), f˜(t) = | R dV =0, (2.1) t=0 t=0 Z (˜g(t),f˜(t)) g+ts dt dt M ˜ where, R(˜g(t),f˜(t)) is the scalar curvature of (˜g(t), f(t)) and ˜ ~ ˜ 2 R(˜g(t),f˜(t)) = Rg˜(t) + △g˜(t)(f(t)) −|∇f(t)| , where, by notation ∇~ we mean the gradient of smooth functions. 2 To find derivation in 2.1, we must compute derivations of Rg˜(t), △g˜(t)(f˜(t)), |∇~ f˜(t)| and dVg+ts for t = 0. In [1], it is shown ′ (Rg˜(t)) (0) = −hs, Rici + div(X), X ∈ X (M), (2.2) 1 (dV )′(0) = hg,sidV . (2.3) g+ts 2 g ˜ ˜ 2 By local computations, we find derivation of △g˜(t)(f(t)) and |∇~ f(t)| at t = 0. By means of a local coordinate system on M with local frame ∂i, we can write ˜ ˜ d ~ ˜ 2 ij ∂f(t) ∂f(t) ′ ij ∂f ∂f ij ∂h ∂f |t=0|∇f(t)| = g˜ (t) i j (0) = −s i j +2g i j dt ∂x ∂x ∂x ∂x ∂x ∂x = −hs, df ⊗ dfi +2h∇~ h, ∇~ fi. Local computation of Laplacian of a smooth function f is as follows. ∂2f ∂f △(f)= gij( − Γk ). ∂xi∂xj ij ∂xk So, 2 ′ ˜ ′ ij ∂ (f + th) ˜k ∂(f + th) [△g˜(t)(f(t))] (0) = g˜ (t)( − Γij(t) ) (0) ∂xi∂xj ∂xk ∂2f ∂f ∂2h ∂f ∂h = −sij( − Γk )+ gij( − Ak − Γk ) ∂xi∂xj ij ∂xk ∂xi∂xj ij ∂xk ij ∂xk ∂f = −hs, Hess(f)i + △(f) − gijAk , (2.4) ij ∂xk k ˜k ′ k ij k where, Aij =(Γij) (0). Define a vector field Y by components Y = g Aij. We can write 1 Y k = gijAk = gijgkl(∇ s + ∇ s −∇ s ) ij 2 i jl j il l ij 1 = gkl(gij∇ s + gij∇ s − gij∇ s ) 2 i jl j il l ij 1 1 = gkl(div(s) + div(s) −∇ tr(s)) = div(s)k − (∇~ tr(s))k. 2 l l l 2 4 Consequently, ∂f ∂f 1 ∂f 1 gijAk = div(s)k − (∇~ tr(s))k = div(s)(∇~ f) − h∇~ tr(s), ∇~ fi. ij ∂xk ∂xk 2 ∂xk 2 In the following, whenever it is convenient, we consider s as a (1, 1) symmetric tensor.
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